#6468158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,5)(3,7,8,6)(9,10,13,16)(11,14,12,15)(17,18)(19,29)(20,35)(21,26)(22,27)(23,38)(24,28)(25,33)(30,46)(31,44)(32,40)(34,45)(36,43)(37,42)(39,48)(41,47);(2,5)(6,7)(11,12)(14,15)(17,19,22,32,24,34,23,33)(18,25,38,45,28,40,27,29)(20,31,37,47,21,30,36,48)(26,41,42,44,35,39,43,46);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,14,16,15)(17,20,24,21)(18,26,28,35)(19,30,34,31)(22,37,23,36)(25,39,40,41)(27,43,38,42)(29,44,45,46)(32,48,33,47);(9,12,13,11)(10,15,16,14)(17,21,24,20)(18,26,28,35)(19,31,34,30)(22,36,23,37)(25,39,40,41)(27,43,38,42)(29,44,45,46)(32,47,33,48);(17,22,24,23)(18,27,28,38)(19,32,34,33)(20,36,21,37)(25,29,40,45)(26,42,35,43)(30,47,31,48)(39,46,41,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=ghi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375354);